THE SHRINKING PROJECTION METHOD FOR GENERALIZED MIXED EQUILIBRIUM PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

Abstract

In this paper, we introduce a new hybrid projection iterative scheme based on the shrinking projection method for two asymptotically quasi-\phi-nonexpansive mappings, for finding a common element of the set of solutions of the generalized mixed equilibrium problems and the set of common fixed points of two asymptotically quasi-\phi-nonexpansive mappings in Banach spaces. The results obtained in this paper improve and extend the recent ones announced by Matsushita and Takahashi [ S. Matsushita, W. Takahashi, Weak and strong convergence theorems for relatively nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. (2004), 2004, 37-47], Qin et al. [X. Qin, S.Y. Cho, S.M Kang, On hybrid projection methods for asymptotically quasi-\phi-nonexpansive mappings, Applied Mathematics and Computation 215 (2010) 38743883], and Chang, Lee and Chan [S.-s. Chang, H.W. Joseph Lee, C.K. Chan, A new hybrid method for solving generalized equilibrium problem variational inequality and common fixed point in Banach spaces with applications, Nonlinear Analysis (2010), doi:10.1016/j.na.2010.06.006] and many others.